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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.

In this page, a simple overview of the divisibility test for 9 is provided.

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Which of the following are multiples of 9?

• 9,18,27,36, cdots
• 9,18,27,36, cdots
• 10,20,30,40,cdots

The answer is "9,18,27,36, cdots"

Consider the multiples of 9 given as 9,18,27,36, cdots. Is there any similarity observed in the multiples?

• all are ending in 9
• sum of all digits is a multiple of 9
• sum of all digits is a multiple of 9

The answer is "sum of all digits is a multiple of 9".

This is explained as follows
18 -> 1+8=9 sum is 9
27 -> 2+7=9 sum is 9
36 -> 3+6=9 sum is 9
189 -> 1+8+9=18 sum is multiple of 9

This is true for all multiples of 9.

Consider the numbers 12 and 81. Which one is divisible by 9?

Check this by long division method.

• 81
• 81
• 12

The answer is "81"

To identify the multiplies of 9, the sum of all digits of the number is checked for divisibility by 9.

Explanation for the curious mind.

Consider divisibility test 42 by 9

To simplify the divisibility test, let us subtract a multiple of the divisor 9.
Seeing the 10s place value 4, we choose the multiple 9xx4 to subtract from the number.

As per the property of simplification by subtraction, the divisibility test of 42 is simplified into divisibility test of 42-36 = 40-36+2 = 4+2.

This explains the divisibility test for 9.

Test for Divisibility by 9 : If the sum of all the digits is divisible by 9, then the number is divisible by 9

Is 2008 divisible by 9?

• Yes
• No
• No

The answer is "No". Checking the sum of digits 2+0+0+8=10, it is concluded that the number is not divisible by 9.

Is 927 divisible by 9?

• Yes
• Yes
• No

The answer is "Yes". Checking the sum of digits 9+2+7=18, it is concluded that the number is divisible by 9.

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